Isotropic Turbulence Model Research Articles

Closure model for homogeneous isotropic turbulence in the Lagrangian specification of the flow field

This paper proposes a new two-point closure model that is compatible with the Kolmogorov$-5/3$power law for homogeneous isotropic turbulence in an incompressible fluid using the Lagrangian specification of the flow field. A closed set of three equations was derived from the Navier–Stokes equation with no adjustable free parameters. The Kolmogorov constant and the skewness of the longitudinal velocity derivative were evaluated to be 1.779 and$-0.49$, respectively, using the proposed model. The bottleneck effect was also reproduced in the near-dissipation range.

Calibration of the Reynolds Stress Model for the Simulation of Gas Flows in Corrugated Tubes

ABSTRACTTubes with corrugated surfaces can clearly increase the overall heat transfer coefficient in comparison to smooth tubes. Due to the additional influence of different curved surfaces, the analytical calculation of the overall heat transfer coefficient and pressure drop fails for those tubes. Numerical simulations using turbulence models with two or three differential equations appropriate for isotropic turbulence are not satisfactory in this case. Since corrugated walls cause anisotropic turbulences in the near wall region, the secondary flow must be considered in the numerical simulation. Within the time averaging turbulence models (Reynolds averaged Navier stokes = RANS models), the Reynolds stress model (RSM) is the only model solving all six components of the Reynolds stress tensor and thus enabling the consideration of secondary and anisotropic turbulence. Most turbulence models are developed and calibrated for smooth wall boundary conditions. As a consequence, simulations of corrugated tubes based on isotropic turbulence models provide inadequate results when compared with measured values. This article describes the calibration of the RSM for the simulation of corrugated tubes using the values of a shell-and-tube test facility for adaptation. To calibrate the heat transfer rate, the dimensionless turbulent Prandtl number is modified. For the pressure drop calculation, the pressure strain coefficient is adjusted varying the appropriate dimensionless coefficient. Moreover, the local flow behavior in the near wall region is validated by local measurements by laser Doppler velocimetry.

Helicity statistics in homogeneous and isotropic turbulence and turbulence models

We study the statistical properties of helicity in direct numerical simulations of fully developed homogeneous and isotropic turbulence and in a class of turbulence shell models. We consider correlation functions based on combinations of vorticity and velocity increments that are not invariant under mirror symmetry. We also study the scaling properties of high-order structure functions based on the moments of the velocity increments projected on a subset of modes with either positive or negative helicity (chirality). We show that mirror symmetry is recovered at small-scales, i.e., chiral terms are subleading and they are well captured by a dimensional argument plus anomalous corrections. These findings are also supported by a high Reynolds numbers study of helical shell models with the same chiral symmetry of Navier-Stokes equations.

Leith diffusion model for homogeneous anisotropic turbulence

Abstract A new spectral closure model for homogeneous anisotropic turbulence is proposed. The systematic development begins by closing the third-order correlation describing nonlinear interactions by an anisotropic generalization of the Leith diffusion model for isotropic turbulence. The correlation tensor is then decomposed into a tensorially isotropic part, or directional anisotropy, and a trace-free remainder, or polarization anisotropy. The directional and polarization components are then decomposed using irreducible representations of the SO(3) symmetry group. Under the ansatz that the decomposition is truncated at quadratic order, evolution equations are derived for the directional and polarization pieces of the correlation tensor. Numerical simulation of the model equations for a freely decaying anisotropic flow illustrate the non-trivial effects of spectral dependencies on the different return-to-isotropy rates of the directional and polarization contributions.

Advances in CFD Modelling of Multiphase Flows in Cyclone Separators

This paper evaluates the predictive performance of modern Computational Fluid Dynamics (CFD) methods in cyclone vortex-separation flows. Computer simulations are performed with the Eulerian-Eulerian (E-E) as well as Eulerian-Lagrangian (E-L) multiphase models, available is STAR-CCM+ CFD solver. Tested momentum closure strategies included typical two-equation isotropic turbulence models as well as high-order Reynolds Stress Models (RSM) and Large Eddy Simulation (LES) technique. The calculated results show satisfactory agreement compared to corresponding Laser Doppler Velocimetry (LDV) measurements and experimental values for the separation efficiency (η).

Development of fatigue life prediction method and effect of 10-minute mean wind speed distribution on fatigue life of small wind turbine composite blade

This study aims to develop a fatigue life prediction method and to identify the effect that a 10-minute mean wind speed distribution has on the fatigue life of a small-scale wind turbine composite blade. First, combining the von Karman isotropic turbulence model and the Weibull distribution for a 10-minute mean wind speed provided us with the 1-Hz full wind history for a specific time period. Accordingly, the fatigue stress spectra at the blade's fatigue-critical locations (FCLs) were created by applying a stress tensor, in which the interaction between flapwise and edgewise bending moments was taken into consideration. The fatigue life of a composite blade can be predicted with a reliability R = 95% by applying the P–S–N curve obtained from the constant amplitude fatigue tests and rainflow cycle counting, and cumulative damage rule to the fatigue stress spectra. To acquire the second-order regression equation, nonlinear regression analysis was performed on the fatigue lives, which were simulated by using the proposed fatigue life prediction method. In this equation, the variables were the shape parameter, K, and the scale parameter, C, of the Weibull distribution for a 10-minute mean wind speed. The effects of the Weibull parameters on fatigue life were evaluated through the sensitivity analysis of the equations.

Wind spectrum and correlation characteristics relative to vehicles moving through cross wind field

In this study, based on Taylor׳s frozen turbulence hypothesis and isotropic turbulence model, the auto-correlation coefficient function of lateral wind fluctuation and cross-correlation coefficient functions of lateral and longitudinal wind fluctuations are derived from the Kaimal spectrum of longitudinal wind fluctuation. Based on the same hypothesis, this study also derived formulations for calculating the power spectra and correlation coefficient functions of wind turbulence relative to moving vehicles. The results can be applied for any vehicle moving direction, and for both longitudinal and lateral components of wind turbulence. The effects of speed ratio of moving vehicle to mean wind velocity and vehicle direction on the characteristics of wind turbulence relative to moving vehicles are investigated. In addition, a closed-form expression is also proposed to approximately represent the square-root coherence function of wind turbulence relative to moving vehicles. It provides useful insights for better understanding the characteristics of turbulent wind relative to moving vehicles, and also provides a theoretical foundation for solving the discontinuity of sudden change of wind fluctuations while the wind fluctuations are simulated using traditional fixed-point spectrum.

A review of turbulence measurements using ground-based wind lidars

Abstract. A review of turbulence measurements using ground-based wind lidars is carried out. Works performed in the last 30 yr, i.e., from 1972–2012 are analyzed. More than 80% of the work has been carried out in the last 15 yr, i.e., from 1997–2012. New algorithms to process the raw lidar data were pioneered in the first 15 yr, i.e., from 1972–1997, when standard techniques could not be used to measure turbulence. Obtaining unfiltered turbulence statistics from the large probe volume of the lidars has been and still remains the most challenging aspect. Until now, most of the processing algorithms that have been developed have shown that by combining an isotropic turbulence model with raw lidar measurements, we can obtain unfiltered statistics. We believe that an anisotropic turbulence model will provide a more realistic measure of turbulence statistics. Future development in algorithms will depend on whether the unfiltered statistics can be obtained without the aid of any turbulence model. With the tremendous growth of the wind energy sector, we expect that lidars will be used for turbulence measurements much more than ever before.

Near-wake characteristics of a model horizontal axis tidal stream turbine

The results of a detailed experimental investigation of the near-wake (up to seven turbine diameters downstream) of a model horizontal axis tidal turbine (HATT) device in a large-scale recirculating water channel facility are reported. An Acoustic Doppler Velocimeter is used to provide detailed three-dimensional mean and turbulent flow field information at five different depths across the full width of the channel downstream of the turbine, giving the most complete three-dimensional velocities and Reynolds normal and shear stress data set yet available. In addition the Reynolds-stress anisotropy tensor is used to illustrate the degree of anisotropy of the Reynolds stress within the turbine's wake. These results reveal the strongly anisotropic nature of the near-wake turbulence suggesting isotropic turbulence models should not be used to model near-wake dynamics. Finally the power-law decay rates of the maximum normalised turbulent kinetic energy differ significantly from those found downstream of grids, meshes or perforated disks, suggesting that previous modelling approaches, which neglected swirl effects and modelled the turbine by absorption discs, may significantly over predict the turbulent kinetic energy decay rate of HATT wakes.

Numerical Benchmark of Nonconventional RANS Turbulence Models for Film and Effusion Cooling

Over the course of the years, several turbulence models specifically developed to improve the predicting capabilities of conventional two-equations Reynolds-averaged Navier–Stokes (RANS) models have been proposed. They have, however, been mainly tested against experiments only comparing with standard isotropic models, in single hole configuration and for very low blowing ratio. A systematic benchmark of the various nonconventional models exploring a wider range of application is hence missing. This paper performs a comparison of three recently proposed models over three different test cases of increasing computational complexity. The chosen test matrix covers a wide range of blowing ratios (0.5–3.0) including both single row and multi-row cases for which experimental data of reference are available. In particular the well-known test by Sinha et al. (1991, “Film-Cooling Effectiveness Downstream of a Single Row of Holes with Variable Density Ratio,” J. Turbomach., 113, pp. 442–449) at BR = 0.5 is used in conjunction with two in-house carried out experiments: a single row film-cooling test at BR = 1.5 and a 15 rows test plate designed to study the interaction between slot and effusion cooling at BR = 3.0. The first two considered models are based on a tensorial definition of the eddy viscosity in which the stream-span position is augmented to overcome the main drawback connected with standard isotropic turbulence models that is the lower lateral spreading of the jet downwards the injection. An anisotropic factor to multiply the off diagonal position is indeed calculated from an algebraic expression of the turbulent Reynolds number developed by Bergeles et al. (1978, “The Turbulent Jet in a Cross Stream at Low Injection Rates: A Three-Dimensional Numerical Treatment,” Numer. Heat Transfer, 1, pp. 217–242) from DNS statistics over a flat plate. This correction could be potentially implemented in the framework of any eddy viscosity model. It was chosen to compare the predictions of such modification applied to two among the most common two-equation turbulence models for film-cooling tests, namely the two-layer (TL) model and the k–ω shear stress transport (SST), firstly proposed and tested in the past respectively by Azzi and Lakeal (2002, “Perspectives in Modeling Film Cooling of Turbine Blades by Transcending Conventional Two-Equation Turbulence Models,” J. Turbomach., 124, pp. 472–484) and Cottin et al. (2011, “Modeling of the Heat Flux For Multi-Hole Cooling Applications,” Proceedings of the ASME Turbo Expo, Paper No. GT2011-46330). The third model, proposed by Holloway et al. (2005, “Computational Study of Jet-in-Crossflow and Film Cooling Using a New Unsteady-Based Turbulence Model,” Proceedings of the ASME Turbo Expo, Paper No. GT2005-68155), involves the unsteady solution of the flow and thermal field to include the short-time response of the stress tensor to rapid strain rates. This model takes advantage of the solution of an additional transport equation for the local effective total stress to trace the strain rate history. The results are presented in terms of adiabatic effectiveness distribution over the plate as well as spanwise averaged profiles.

Attractive fixed-point solution study of shell model for homogeneous isotropic turbulence

The attractive fixed-point solution of a nonlinear cascade model is studied for the homogeneous isotropic turbulence containing a parameter C, introduced by Desnyansky and Novikov. With a traditional constant positive external force added on the first shell equation, it can be found that the attractive fixed-point solution of the model depends on both the parameter C and the external force. Thus, an explicit force is introduced to remove the effects of the external force on the attractive fixed-point solution. Furthermore, two groups of attractive fixed-point solutions are derived theoretically and studied numerically. One of the groups has the same scaling behavior of the velocity in the whole inertial range and agrees well with those observed by Bell and Nelkin for the nonnegative parameters. The other is found to have different scaling behaviors of the velocity at the odd and even number shells for the negative parameters. This special characteristic may be used to study the anomalous scaling behavior of the turbulence.

On Alternative Model of Sound Scattering in Turbulent Moving Media

Model of sound scattering in turbulent mediu m co mprising chaotically distributed in space moving in manifold directions spherically symmet ric structures - localized flo ws, say, vortices of various linear dimensions from s mallest Kolmogorov's to outer turbulent scales is proposed. Scattering crossections and distance attenuation parameters related to structures motion in the presence and absence of vorticity inside them for sound waves are calculated on etalon problem solutions basis. Resulting frequency dependencies and scattering laws in model med iu m differ substantially not only fro m Rayleigh law but fro m several other well known predicted patterns of sound scattering in turbulent mediu m as well. Attenuation value is expected to depend on vortices Mach number to scattering wave parameter ratio. Model parameter control is available by means of dimensions, velocities and concentration of basic localized flo w changes corresponding to turbulence scale, intensity and degree of development changes. Co mparison of expected attenuation parameters with experimental data and estimates based on classical isotropic turbulence models is presented. PACS nu mbers: 43.20.Fn, 43.28.Gq, 43.28.Py

Wavefronts in a Relativistic Two-Phase Turbulent Flow

The simplest model of isotropic relativistic turbulence consists of the relativistic two-phase fluid equations augmented by an equation for the turbulent energy. This model can also be viewed as the relativistic Euler equations for a fluid with three independent entropies. Two of them are the conventional thermodynamic entropies of each phase, and the other one is associated with the turbulent energy. The resulted system of the governing equations is hyperbolic, therefore the weak discontinuities compatible with such system are studied.

Stationarity of linearly forced turbulence in finite domains

A simple scheme of forcing turbulence away from decay was introduced by Lundgren some time ago, the "linear forcing," which amounts to a force term that is linear in the velocity field with a constant coefficient. The evolution of linearly forced turbulence toward a stationary final state, as indicated by direct numerical simulations (DNS), is examined from a theoretical point of view based on symmetry arguments. In order to follow closely the DNS, the flow is assumed to live in a cubic domain with periodic boundary conditions. The simplicity of the linear forcing scheme allows one to rewrite the problem as one of decaying turbulence with a decreasing viscosity. Its late-time behavior can then be studied by scaling symmetry considerations. The evolution of the system in the description of "decaying" turbulence can be understood as the gradual symmetry breaking of a larger approximate symmetry to a smaller symmetry that is exact at late times. The latter symmetry implies a stationary state: In the original description all correlators are constant in time, while, in the "decaying" turbulence description, that state possesses constant Reynolds number and integral length scale. The finiteness of the domain is intimately related to the evolution of the system to a stationary state at late times: In linear forcing there is no other large scale than the domain size, therefore, it is the only scale available to set the magnitude of the necessarily constant integral length scale in the stationary state. A high degree of local isotropy is implied by the late-time exact symmetry, the symmetries of the domain itself, and the solenoidal nature of the velocity field. The fluctuations observed in the DNS for all quantities in the stationary state can be associated with deviations from isotropy that is necessarily broken at the large scale by the finiteness of the domain. Indeed, to strengthen this conclusion somewhat, self-preserving isotropic turbulence models are used to study evolution from a direct dynamical point of view. Simultaneously, the naturalness of the Taylor microscale as a self-similarity scale in this system is emphasized. In this context the stationary state emerges as a stable fixed point. We also note that self-preservation seems to be the reason behind a noted similarity of the third-order structure function between the linearly forced and freely decaying turbulence, where, again, the finiteness of the domain plays a significant role.

Dynamics of charged particles and perpendicular diffusion in turbulent magnetic field

A test particle code is employed to explore the dynamics of charged particles and perpendicular diffusion in turbulent magnetic field, where a three-dimensional (3D) isotropic turbulence model is used in this paper. The obtained perpendicular diffusion at different particle energies is compared with that of the nonlinear guiding center (NLGC) theory. It is found that the NLGC theory is consistent with test particle simulations when the particle energies are small. However, the difference between the NLGC theory and test particle simulations tends to increase when the particle energy is sufficiently large, and the threshold is related to the turbulence bend-over length. In the NLGC theory, the gyrocenter of a charged particle is assumed to follow the magnetic field line. Therefore, when the particle has sufficiently large energy, its gyroradius will be larger than the turbulence bend-over length. Then the particle can cross the magnetic field lines, and the difference between the test particle simulations and NLGC theory occurs.

Numerical modelling of the carrier gas phase in a laboratory-scale coal classifier model

The pneumatic transport of fine ideally combustible coal dust to the burner furnace is an important process in coal fired power plants. The strongly swirling air phase responsible for the particle separation and transport in a coal pulverising mill was characterised experimentally and numerically. Measurements of the swirl velocity component were taken in a scaled laboratory model of the device and compared to CFD model. In particular, an evaluation of the turbulence models used to describe the flow was performed. The modified isotropic k-epsilon turbulence models (RNG k-ε and Realizable k-ε) were compared to the anisotropic Reynolds stress model (RSM) and their ability to predict the bulk flow structure present in the classifier was assessed. The experiments showed that the swirling flow structure, responsible for coarse-fine particle classification, has several flow regimes which are governed by the areas it is bounded by. The numerical model predictions generally corroborate the results. However, a distinction in performance between the three models can be made based on accuracy, solution generation time and numerical stability. The RSM model predicted both the trends and magnitude the most accurately when compared to the isotropic models. However, the Realizable k-epsilon model, with its relatively low solution generation time, shows potential when using CFD as a classifier design optimization tool. The investigation has given some insight on single phase classifier flow and suggests a design improvement based on the results.

New development of the turbulent Prandtl number models for the computation of film cooling effectiveness

This paper presents the new development on the turbulent Prandtl number distribution models which are used to improve the computation accuracy of film cooling effectiveness distribution in the frame of the isotropic turbulence models. In the new developed turbulent Prandtl number distribution models, the turbulent Prandtl number varies with some characteristic quantity of the film cooling to make the models applicable for more film cooling cases effectively. A new VTG-Pr t model is developed for the computation of film cooling cases in which the wall is mainly influenced by the mixing layer region between the jet core and the mainstream. The Pr t value in this model varies with dimensionless temperature gradient which is suggested to be the characteristic quantity of the mixing layer region. The VTG-Pr t model has been used in the computations of long cylindrical hole film cooling, short cylindrical hole film cooling, compound angle cylindrical hole film cooling and expansion shaped hole film cooling in the present paper. And it is validated to be an effective and easy way to obtain accurate film cooling effectiveness distributions for this kind of film cooling. But for the kind of film cooling, in which the wall is mainly influenced by the jet core region and the influence of heat diffusion in the mixing layer region on the cooling effectiveness is very small, the VTG-Pr t model is disabled. Another turbulent Prandtl number distribution model, named VT-Pr t model, is developed for this kind of film cooling in the present paper. The Pr t value in this model varies with dimensionless temperature which is suggested to be the characteristic quantity of the jet core region. Good agreement between the computed and measured results is obtained through the use of VT-Pr t model in the computation of converging slot hole film cooling.

On Corrsin equation closure

Corrsin equation closure is done using the gradient hypothesis relating a two-point third-order correlation moment to a two-point second-order correlation function of a passive scalar field. A numerical locally isotropic turbulence model based on a closed system of Kolmogorov and Yaglom equations is constructed. A similarity solution of the Corrsin equation, which corresponds to infinitely high Reynolds and Peclet numbers, is constructed under assumption of constancy of Corrsin and Loitsiansky invariants. A numerical model of turbulence dynamics and temperature fluctuations behind a heated grid in a wind tunnel, which is based on Karman-Howarth and Corrsin closed equations, is developed.

Application of the TLVA model for predicting film cooling under rotating frames

An in-house three-dimensional Navier–Stokes code was used to evaluate the advantages of anisotropic turbulence models over the classical isotropic turbulence models for the prediction of film cooling under rotating conditions. The anisotropic turbulence model we chose was the two-layer TLVA model and the isotropic turbulence models were the standard k–ε, the k–ω and the SST models. For the purpose of validation of numerical results, a test rig was setup and experiments were carried out for film cooling under rotation. The test model had a flat test surface with a 4 mm diameter straight circular cooling hole in 30° inclined injection and it rotated at four different speeds of 0, 500, 800 and 1000 rpm. Experiments were accomplished with the momentum ratio set to be 0.285, the Reynolds number kept at 1.45 × 10 5 and the averaged density ratio at 1.026. Comparison indicated that the TLVA anisotropic turbulence model preformed best against its isotropic counterparts and produced the closest local cooling effectiveness η to the experimental results of all conditions. Detailed flow and temperature field analysis revealed that the improvement of anisotropic turbulence model was mostly due to its ability in accurately simulating the film lateral spreading. On the contrary, the isotropic turbulence models heavily underestimated the lateral spreading of the cooling film and this led to the overshooting of cooling effectiveness along the centerline regions and the undershooting for the rest parts. Apart from the cooling effectiveness, deflection of the cooling film from centerline due to Coriolis and centrifugal forces under coordinate rotation was also best predicted by the TLVA model.

Evaluation of linear and nonlinear convection schemes on multidimensional non‐orthogonal grids with applications to KVLCC2 tanker

AbstractAlmost all evaluations of convection schemes reported in the literature are conducted using simple problems on uniform orthogonal grids; thus, having limited contribution when solving industrial computational fluid dynamics (CFD), where the grids are usually non‐orthogonal with distortions. Herein, several convection schemes are assessed in uniform and distorted non‐orthogonal grids with emphasis on industrial applications. Linear and nonlinear (TVD) convection schemes are assessed on analytical benchmarks in both uniform and distorted grids. To evaluate the performance of the schemes, four error metrics are used: dissipation, phase and L1 errors, and the schemes' effective order of accuracy. Qualitative and quantitative deterioration of these error metrics as a function of the grid distortion metrics are investigated, and rigorous verifications are performed. Recommendations for effective use of the convection schemes based on the range of grid aspect ratio (AR), expansion ratio (ER) and skewness (Q) are included. A ship hydrodynamics case is studied, involving a Reynolds averaged Navier–Stokes simulation of a bare‐hull KVLCC2 tanker using linear and nonlinear convection schemes coupled with isotropic and anisotropic Reynolds‐stress (ARS) turbulence models using CFDShip‐Iowa v4. Predictions of local velocities and turbulent quantities from the midships to the nominal wake plane are compared with experimental fluid dynamics (EFD), and rigorous verification and validation analyses for integral forces and moments are performed for 0° and 12° drift angles. Best predictions are observed when coupling a second‐order TVD scheme with the anisotropic turbulence model. Further improvements are observed in terms of prediction of the vortical structures for 30° drift when using TVD2S‐ARS coupled with DES. Copyright © 2009 John Wiley & Sons, Ltd.